SSC CGL 2025 (Held On: 16 Sept, 2025 Shift 3) (Previous Year Paper)

Correct Option - 2

Given: Italy is a beautiful village and China is a big building.

Words in the sentence that begin with a consonant: Italy is a beautiful village and China is a big building

Thus, there are 5 words that begin with a consonant.

Hence, the correct answer is "Option 2"

Correct Option - 2

The position and positional values or English Alphabets are:

Given: GBD, JFJ, MJP, PNV ?

 

The logic followed here is:

Thus, SRB will complete the series.

Hence, the correct answer is "Option 2".

Correct Option - 4

 

Thus, option 4 is exactly the same as the one given.

Hence, the correct answer is "Option 4".

Correct Option - 1

The position and positional values or English Alphabets are:

Given: SCEG, S F K P, S I Q Y, S L W H, ?

 

The logic followed here is:

Thus, SOCQ will complete the series.

Hence, the correct answer is "Option 1".

Correct Option - 1

Statement: C= A ≤ T < D ≥ O; G ≥ D < S; T ≥ A

Conclusions:

I: A < O → False (As, A ≤ T < D ≥ O, there is no direct relationship between A and O)

II: C < G → True (C= A ≤ T < D ≤ G, it gives C < G)

Thus, only Conclusion II follows.

Hence, the correct answer is "Option 1".

Correct Option - 3

Given: 2, 12, 36, 80, 150, ?

The logic followed here is:

Thus, 252 will complete the pattern.

Hence, the correct answer is "Option 3".

Correct Option - 1

The logic followed here is:

Logic: a # b = a ÷ b

1. 10 # 5 = 5 ⇒ 10 ÷ 5 ≠ 2
2. 20 # 4 = 5 ⇒ 20 ÷ 4 = 5
3. 25 # 5 = 5 ⇒ 25 ÷ 5 = 5
4. 15 # 3 = 5 ⇒ 15 ÷ 3 = 5

Thus, 10 # 5 = 5 is the odd one out.

Hence, the correct answer is "Option 1".

Given:

Statement: The rapid advancement in gene editing technologies such as CRISPR offers immense potential for curing genetic disorders, but it also raises concerns about the ethical implications of editing the human genome.

Assumptions:

I. Gene editing technologies can potentially cure genetic disorders. - Implicit (The statement clearly implies these technologies offer immense potential for curing genetic disorders. If this were not true, the statement would collapse)

II. Ethical concerns are a natural consequence of advancing gene editing technologies - Implicit (The statement implies advancement raises ethical concerns. For this to be valid, we must assume that such technological progress leads to ethical issues)

III. The potential for gene editing to cure genetic disorders outweighs the ethical concerns - Not Implicit (The statement does not compare which is greater. It simply mentions both potential and concerns — no judgment is made)

So, only I and II are implicit.

Hence, the correct answer is "Option 1".

Given:

Statement: The bank has introduced a biometric login system.

Now,

Conclusions:

I. The bank wants to improve security. - True (Because biometric login systems are generally introduced to enhance security and reduce fraud. So it is reasonable to conclude that the bank’s intention is to improve security.)

II. Customers must carry biometric IDs. - False (Because biometric login usually uses fingerprints, face recognition, etc. — it does not necessarily require customers to carry separate biometric IDs.)

So, only Conclusion I follows.

Hence, the correct answer is "Option 1".

Correct Option - 3

The position and positional values or English Alphabets are:

The logic followed here is: Letter are coded as their opposite letters.

ACFL : ZXUM

Similarly

Thus, BDEM : YWVN

Hence, the correct answer is "Option 3".

Correct option - 4

Given Statements:

Some animals are carnivores. Some carnivores are wild.

The least possible Venn diagram for the given statement is:

Conclusions:

I. Some animals are wild. - False (Because, no definite relation given between animals and wild, so it can be possible).

II. All wild are animals. - False (Because, no definite relation given between animals and wild, so it can be possible).

So, neither I nor II follows.

Hence, the correct answer is "Option 4".

Correct Option - 1

Given Word: QUARRELSOME

1. AIR → QUARRELSOME → Cannot be formed as 'I' is missing 2. SOME → QUARRELSOME 3. QUARREL → QUARRELSOME 4. LOAM → QUARRELSOME

So, AIR is the word which cannot be formed using the letters of the given word.

Hence, the correct answer is "Option 1".

Given:

6th term = 48

First term (a) = 1.5

Formula Used:

n^th term of GP = a × r^(n-1)

Calculation:

⇒ 6th term = a × r^(6-1)

⇒ 48 = 1.5 × r⁵

⇒ r⁵ = 48 / 1.5

⇒ r⁵ = 32

⇒ r = 32^1/5

⇒ r = 2.0

∴ The correct answer is 2.0.

Correct Option - 4

1. Bhagat Singh → Bhagat Singh is a well known freedom fighter of India.
2. Chandrashekhar Azad → Chandrashekhar Azad is a well known freedom fighter of India. 3. Rajguru → Rajguru is a well known freedom fighter of India. 4. Manmohan Singh → Manmohan Singh is a former Prime Minister of India.

All options except Manmohan Singh are freedom fighters of India.

Hence, the correct answer is "Option 4".

Correct Option - 2

Family Chart:

Given: P is the mother of Q. Q is the sister of R.

P is mother of R.

Hence, the correct answer is "Option 2".

Correct Option - 2

Given: 

PLANE is coded as 74851.

DOG is coded as 392.

Similarly,

Thus, GOLD is coded as 2943.

Hence, the correct answer is "Option 2".

Given:

√144

³√27

Formula Used:

√x = x^1/2

³√x = x^1/3

Calculation:

⇒ √144 = 144^1/2 = 12

⇒ ³√27 = 27^1/3 = 3

⇒ √144 + ³√27 = 12 + 3 = 15

∴ The correct answer is 15.

Correct Option - 4

Given: If 8 # 2 = 64, 7 # 3 = 343

The logic followed here is:

Logic: a # b = a^b

⇒ 8 # 2 = 64 ⇒ 82 = 64 

⇒ 7 # 3 = 343 ⇒ 73 = 343

Similarly,

⇒ 6 # 2 = 62 = 36

Hence, the correct answer is "Option 4".

Correct Option - 1

The position and positional values or English Alphabets are:

The logic followed here is:

 

Similarly,

So, CHAIR will be written as DKBJS.

Hence, the correct answer is "Option 1".

Given:

Cost of each eraser = ₹8

Cost of each folder = ₹25

Number of erasers = 5

Number of folders = 3

Formula Used:

Total bill = (Number of erasers × Cost of each eraser) + (Number of folders × Cost of each folder)

Calculation:

⇒ Total bill = (5 × ₹8) + (3 × ₹25)

⇒ Total bill = ₹40 + ₹75

⇒ Total bill = ₹115

∴ The correct answer is ₹115.

Correct Option - 3

Family Chart:

Given: P ÷ Q - R

It means: P is father of Q, Q is sister of R

Clearly, P is R's father.

Hence, the correct answer is "Option 3".

Correct Option - 2

BODMAS Table:

1. 12 - 4 × 2 + 1 = 7

After replacing the symbols by their meaning, we get:

⇒ 12 + 4 - 2 ÷ 1 = 7

⇒ 12 + 4 - 2 = 7 

⇒ 16 - 2 = 7 

⇒ 14 ≠ 7

2. 10 + 2 - 4 × 1 = 8

After replacing the symbols by their meaning, we get:

⇒ 10 ÷ 2 + 4 - 1 = 8

⇒ 5 + 4 - 1 = 8

⇒ 9 - 1 = 8

⇒ 8 = 8

3. 16 ÷ 4 + 2 - 2 = 10

After replacing the symbols by their meaning, we get:

⇒ 16 × 4 ÷ 2 + 2 = 10

⇒ 16 × 2 + 2 = 10

⇒ 32 + 2 = 10

⇒ 34 ≠ 10

4. 8 × 2 - 2 + 2 = 1

After replacing the symbols by their meaning, we get:

⇒ 8 - 2 ÷ 2 + 2 = 1

⇒ 8 - 1 + 2 = 1

⇒ 10 - 1 = 1

⇒ 9 ≠ 1

Hence, the correct answer is "Option 2".

Correct Option - 1

BODMAS Table:

Given: 25 B 5 C 3 D 2 A 4 = ?

After replacing the letters by their meaning, we get:

⇒ 25 ÷ 5 + 3 × 2 - 4 = ?

⇒ 5 + 3 × 2 - 4 = ?

⇒ 5 + 6 - 4 = ?

⇒ 11 - 4 = ?

⇒ 7 = ?

Hence, the correct answer is "Option 1".

Correct Option - 1

BODMAS Table:

Given equation: 5 - 2 x 3 + 1

After replacing the symbols by their meaning, we get:

= 5 ^ 2 - 3 × 1

=  5² - 3 × 1

= 25 - 3 × 1

= 25 - 3

= 22

Hence, the correct answer is "Option 1".

Given:

Sum of numbers: 11 + 40

Multiplier: 5

Formula Used:

Sum = Number 1 + Number 2

Result = Sum × Multiplier

Calculation:

⇒ Sum = 11 + 40

⇒ Sum = 51

⇒ Result = 51 × 5

⇒ Result = 255

∴ The correct answer is 255.

The correct answer is Option 1.

Key Points

• Perini Shivatandavam is a warrior dance that was traditionally performed before soldiers set off for war.
• The dance form has its origins in the Kakatiya dynasty, which ruled parts of present-day Telangana and Andhra Pradesh during the 12th and 13th centuries.
• It was revived in modern times by the renowned dance guru Nataraja Ramakrishna.
• The dance is dedicated to Lord Shiva, and its performance is characterized by vigorous movements symbolizing strength and valor.
• The second statement, which associates the dance form with the Chola kings of Tamil Nadu, is incorrect as Perini Shivatandavam is specific to the Kakatiya dynasty of Telangana.

Additional Information

• Kakatiya Dynasty:
• The Kakatiya dynasty ruled the Deccan region, particularly Telangana and parts of Andhra Pradesh, during the 12th and 13th centuries.
• They were known for their patronage of art and architecture, including temples like the Ramappa Temple, a UNESCO World Heritage Site.
• Chola Dynasty:
• The Chola dynasty ruled over Tamil Nadu and parts of South India, Sri Lanka, and the Maldives during the medieval period.
• They were known for their contributions to art, literature, and architecture, including the construction of the Brihadeeswarar Temple.

The correct answer is Flat arched entrance with pediment.

Key Points

• The Capuchin Church architecture is characterized by its simplicity and austerity, reflecting the principles of the Capuchin Order.
• A prominent feature is the flat arched entrance with a pediment, which aligns with the minimalist design intended to avoid excessive ornamentation.
• These churches emphasize functionality over decorative elements, adhering to the Capuchin’s spiritual values of poverty and humility.
• The pediment, a triangular or curved gable, often sits atop the entrance, adding to the balanced and modest architectural style.
• The interiors of Capuchin churches are usually unadorned, focusing on creating a serene atmosphere for prayer and reflection.

Additional Information

• Ornate spires and domes:
• This feature is typical of Gothic and Baroque architecture but is not associated with Capuchin churches.
• Ornate spires are tall, pointed structures that symbolize a connection to the heavens, commonly found in European cathedrals.
• Domes, often lavishly decorated, are a hallmark of Renaissance and Baroque church designs, seen in structures like St. Peter's Basilica in the Vatican.
• Complex traceried rose windows:
• These are intricate circular stained-glass windows, typically seen in Gothic cathedrals like Notre Dame in Paris.
• They are known for their elaborate designs and colorful depictions of biblical stories, which are not in line with the plain aesthetics of Capuchin churches.
• Multi-storey shikhara tower:
• A shikhara is the rising tower typically found in Hindu temple architecture, specifically in Northern India.
• Multi-storey shikharas are designed to symbolize Mount Meru, a sacred mountain in Hindu cosmology, and are not a feature of Christian or Capuchin church architecture.

The correct answer is The name "Teej" comes from a red insect seen in monsoon.

Key Points

• The word "Teej" is derived from a tiny red insect (locally known as "Teej") that emerges during the monsoon season, symbolizing fertility and the arrival of rain.
• The insect signifies the onset of the monsoon, which is a crucial time for agriculture and fertility, aligning with the festival's themes of renewal and prosperity.
• Teej is widely celebrated in Northern India and Nepal, marking the monsoon season and honoring nature's bounty.
• The festival is deeply rooted in Indian culture, where mythology and seasonal changes are intertwined to emphasize the harmonious relationship between humans and nature.
• The red insect also symbolizes the vibrant and colorful rituals of Teej, such as women adorning themselves in red and green attire, which represent fertility and prosperity.

Additional Information

• Major Pan-Indian Festivals (Widely Celebrated)
• Diwali: Festival of lights, symbolizing victory of light over darkness (Oct/Nov).
• Holi: Spring festival of colors, celebrating joy and new beginnings (Mar).
• Dussehra/Vijayadashami: Commemorates Lord Rama's victory over Ravana (Oct/Nov).
• Navratri/Durga Puja: Nine nights dedicated to Goddess Durga, signifying divine feminine power.
• Eid-ul-Fitr: Marks the end of Ramadan for Muslims.
• Janmashtami: Celebrates the birth of Lord Krishna (Aug/Sep).

The correct answer is He acted as the foreign minister and collaborated closely with the king and military.

Key Points

• The Mahasandhivigrahika was an important official during the Gupta period and acted as the foreign minister.
• This officer was responsible for maintaining diplomatic relations and treaties with other kingdoms and empires.
• The Mahasandhivigrahika worked closely with the king and the military in strategic planning, particularly during times of war and peace negotiations.
• He played a crucial role in the administration by managing alliances, resolving disputes with neighboring states, and ensuring the stability of the kingdom's external relations.
• The title "Mahasandhivigrahika" itself translates to "the great officer of war and peace," highlighting his prominent role in diplomacy and military affairs.

Additional Information

• Administrative structure of the Gupta Empire
• The Gupta Empire (circa 320–550 CE) had a centralized administrative system with various officials handling specific responsibilities such as taxation, justice, military, and diplomacy.
• The Mahasandhivigrahika was one of the high-ranking officials, reflecting the advanced and organized governance of the Gupta Empire.
• The Gupta period is often referred to as the "Golden Age" of India due to advancements in science, literature, and administration.

The correct answer is Hafiz Nagma - Dance based on musical notes of Sufiana music.

Key Points

• Hafiz Nagma is a traditional dance form performed in Kashmir, with its roots deeply embedded in Sufiana music.
• It is based on musical notes and rhythms of Sufiana music, often performed by women dancers called “Hafizas.”
• The dance is accompanied by classical Kashmiri instruments such as the Santoor, Sitar, and Tabla.
• The dance movements are intricately aligned with the melody and spiritual essence of Sufiana music, creating a harmonious and soulful experience.
• This art form not only represents Kashmiri culture but also highlights the fusion of music and dance in Islamic traditions.

Additional Information

• Chakri
• Chakri is a popular folk music form of Kashmir, often used for storytelling.
• This music is performed using traditional instruments like the Rabab, Harmonium, and Sarangi.
• It is not a classical form or associated with the Santoor and Hafiz Nagma; instead, it is a lighter, conversational style of music.
• Sufiana Music
• Sufiana music is a classical music tradition of Kashmir inspired by Persian and Central Asian influences.
• It is performed using instruments like the Santoor, Setar, and Tabla, and is known for its spiritual and meditative qualities.
• This music is not a folk style and does not include instruments such as Garaha and Rabab as mentioned in the question.
• Rouf
• Rouf is a traditional group dance performed by Kashmiri women, especially during festivals like Eid and other celebrations.
• It is characterized by synchronized foot movements and singing of traditional Kashmiri songs.
• It is not an instrumental duet and does not involve the use of Santoor and Sitar.

The correct answer is 3.

Key Points

• In pole vault competitions, a participant is allowed up to three consecutive failed attempts before being disqualified.
• A failed attempt occurs when the vaulter either knocks down the crossbar or does not complete the jump successfully within the allotted time.
• The three failed attempts can be at the same height or across multiple height levels during the competition.
• Once a vaulter fails three attempts consecutively, they are out of the competition, regardless of the heights attempted.
• This rule ensures fairness and aligns with the regulations set by the International Association of Athletics Federations (IAAF) (now World Athletics).

Additional Information

• Pole Vault Rules:
• Pole vault is a track and field event where athletes use a pole to propel themselves over a high bar.
• Each vaulter is given a specific amount of time (usually 1 minute) to execute their jump once their turn begins.
• The height of the bar progressively increases during the competition, and athletes must clear the bar without knocking it off.
• All rules, including the number of failed attempts, are governed by the World Athletics standards for official competitions.

The correct answer is Option 1: A → 3, B → 4, C → 1, D → 2.

Key Points

• Gurjara-Pratiharas: This dynasty primarily ruled over regions corresponding to present-day Gujarat and Rajasthan, which were significant centers of trade and culture during their reign.
• Rashtrakutas: They dominated the Western Deccan region, which included parts of Maharashtra and Karnataka. The Rashtrakutas were known for their contributions to art and architecture, especially the rock-cut temples at Ellora.
• Palas: The Pala dynasty ruled over Bengal and parts of Bihar. They were patrons of Buddhism and contributed immensely to the spread of the religion in Southeast Asia.
• Cholas: The Chola dynasty was centered in Tamil Nadu, known for its maritime trade and naval dominance. They were renowned for their temple architecture and cultural contributions, including the Brihadeeswarar Temple.
• The above matches align with historical and archaeological evidence regarding the geographical dominance of these dynasties.

Additional Information

• Gurjara-Pratiharas:
• The Gurjara-Pratiharas were instrumental in resisting Arab invasions into India during the 8th century.
• Their empire extended over North India, but Gujarat and Rajasthan were their core regions.
• They were known for their patronage of Hindu temple architecture.
• Rashtrakutas:
• Rashtrakutas emerged as one of the most powerful dynasties in peninsular India during the medieval period.
• Their reign is marked by the construction of the Kailasanatha Temple at Ellora, a UNESCO World Heritage Site.
• They were known for their contributions to literature in Kannada and Sanskrit.
• Palas:
• The Palas ruled Bengal and Bihar, and their empire flourished between the 8th and 12th centuries CE.
• They were staunch supporters of Mahayana Buddhism and established several monasteries and universities, including Nalanda and Vikramashila.
• The Pala art and architecture influenced regions beyond India, particularly in Tibet and Southeast Asia.
• Cholas:
• The Cholas were known for their administrative efficiency and naval power, which facilitated their overseas expeditions to Southeast Asia.
• Their reign saw the construction of monumental temples like the Brihadeeswarar Temple, which is a UNESCO World Heritage Site.
• They fostered the growth of Tamil literature and culture, leaving a lasting impact on the region.

The correct answer is Glasgow.

Key Points

• Glasgow is set to host the 2026 Commonwealth Games after Australia withdrew from hosting due to cost concerns and logistical issues.
• The Commonwealth Games Federation (CGF) announced Glasgow as the new host due to its prior experience in hosting major international sporting events, including the 2014 Commonwealth Games.
• Glasgow has a well-developed infrastructure, including sports facilities and transportation networks, suitable for hosting an event of this magnitude.
• The city is expected to leverage its existing venues, such as the Emirates Arena and Hampden Park, to minimize additional expenditure.
• The decision to allocate the Games to Glasgow aligns with the CGF's efforts to ensure sustainability and financial viability for future host cities.

Additional Information

• Birmingham:
• Birmingham hosted the 2022 Commonwealth Games, which featured 19 sports and drew participants from 72 nations and territories.
• The event was highly praised for its inclusivity, as it became the first Commonwealth Games to feature more medal events for women than men.
• The city's facilities, such as the Alexander Stadium, underwent significant upgrades to host the event.
• Victoria:
• Victoria, a state in Australia, was initially set to host the 2026 Commonwealth Games but withdrew in 2023 due to escalating costs that exceeded its budgetary expectations.
• The decision to withdraw highlighted the financial challenges associated with hosting such large-scale international events.
• Toronto:
• Toronto, Canada, has not hosted the Commonwealth Games but is a prominent city known for hosting international sports events, such as the 2015 Pan American Games.
• The city has a strong infrastructure for sports and cultural events, making it a potential candidate for future Games.

The correct answer is North-South Transport Corridor.

Key Points

• The North-South Transport Corridor (NSTC) is a multimodal transportation network aimed at enhancing connectivity between India, Russia, Iran, and Central Asia.
• The corridor includes the use of Chabahar Port in Iran, which serves as a crucial gateway for India to access Central Asia, bypassing Pakistan.
• It was officially established in 2000 by India, Iran, and Russia and has since expanded to include several other countries.
• The NSTC reduces transportation time and costs for goods traveling between India and Central Asia, Europe, and Russia via a combination of sea, rail, and road networks.
• The strategic use of Chabahar Port is part of India's broader geopolitical and economic strategy to counter China's Belt and Road Initiative (BRI) and strengthen India's trade and energy ties with Central Asia.

Additional Information

• East-West Corridor
• The East-West Corridor is a part of India’s Golden Quadrilateral project aimed at improving road connectivity across the country.
• This corridor connects Silchar in Assam (Eastern India) to Porbandar in Gujarat (Western India).
• It primarily focuses on domestic connectivity and has no direct role in enhancing India’s access to Central Asia.
• India-Myanmar Highway
• The India-Myanmar-Thailand Trilateral Highway is an important infrastructure project aimed at improving connectivity between India and Southeast Asia.
• This highway connects Moreh in Manipur (India) to Mae Sot in Thailand via Myanmar.
• While significant for India's Act East Policy, it does not facilitate access to Central Asia.
• South Asian Maritime Path
• This term generally refers to maritime routes used for trade within South Asia and beyond, including the Indian Ocean Region.
• It is not a formal or specific corridor and does not pertain to Central Asian connectivity through Chabahar Port.

The correct answer is Complex mother-daughter bond.

Key Points

• Burnt Sugar, written by Avni Doshi, explores the deeply intricate and often tumultuous relationship between a mother and her daughter.
• The novel is centered on Tara, a rebellious mother, and Antara, her daughter, who struggles with resentment and unresolved emotions stemming from her childhood.
• As Tara begins to suffer from memory loss, Antara is faced with the challenge of caring for the very person who once neglected her.
• The narrative delves into themes of memory, trauma, and the cyclical nature of familial relationships, making it a profound exploration of emotional complexities.
• Burnt Sugar was shortlisted for the Booker Prize in 2020, gaining critical acclaim for its raw portrayal of maternal bonds and its lyrical writing style.

Additional Information

• Key Indian Booker Prize Winners
• Arundhati Roy (1997): First Indian citizen to win the Booker Prize for Fiction for her debut novel, The God of Small Things, a poignant story set in Kerala.
• Salman Rushdie (1981): Won the Booker Prize for Midnight's Children, a magical realist novel about India's transition to independence.
• Kiran Desai (2006): Awarded the Booker Prize for The Inheritance of Loss, exploring themes of globalisation and migration.
• Aravind Adiga (2008): Won the Booker Prize for The White Tiger, a satirical novel about class and corruption in modern India.

The correct answer is Ek Ped Maa Ke Naam 2.0 tree-planting.

Key Points

• On World Environment Day 2025, Education Minister Dharmendra Pradhan promoted the Ek Ped Maa Ke Naam 2.0 campaign to encourage tree plantation as a tribute to mothers.
• The campaign emphasizes environmental sustainability and fostering a culture of gratitude by dedicating trees to mothers.
• It is a continuation of the earlier Ek Ped Maa Ke Naam initiative, which gained significant public participation in its first phase.
• The initiative aligns with the government’s broader focus on combating climate change and conserving biodiversity.
• Under this campaign, citizens are encouraged to plant trees in their homes, schools, and public spaces while promoting awareness about their environmental benefits.

Additional Information

• Plastic-Free India Campaign:
• This campaign focuses on eliminating single-use plastics and promoting sustainable alternatives to reduce plastic pollution.
• India has pledged to phase out single-use plastics by 2025 and aims to increase public awareness through initiatives like Swachh Bharat Abhiyan.
• The campaign has seen active participation from industries, educational institutions, and NGOs to reduce plastic waste and promote recycling.
• Clean Ganga Youth Movement:
• This initiative is part of the larger Namami Gange Programme, aiming to rejuvenate and conserve the Ganga river.
• It engages youth in activities like cleaning riverbanks, preventing pollution, and spreading awareness about the importance of clean water bodies.
• The movement highlights the role of young citizens in environmental conservation and water resource management.
• National Coastal Cleanup Drive:
• This program focuses on cleaning coastal areas and addressing marine pollution caused by human activities.
• It involves volunteers, local communities, and environmental organizations to remove waste from beaches and restore marine ecosystems.
• The drive is crucial for protecting marine biodiversity and mitigating the impacts of ocean pollution.

The correct answer is 31%.

Key Points

• India's urban population percentage in the 2011 Census was 31.16%, which is commonly rounded off to 31%.
• The Census 2011 data revealed that the urban population in India increased significantly compared to previous decades, reflecting urbanization trends.
• In absolute terms, the urban population of India in 2011 was approximately 377 million, out of a total population of 1.21 billion.
• This urbanization was driven by migration, natural population growth, and the reclassification of rural areas into urban areas.
• States like Maharashtra, Uttar Pradesh, and Tamil Nadu had the highest urban populations in absolute terms in the 2011 Census.

Additional Information

• Key Population Figures
• Total Population: 1,210,854,977 (1.21 billion).
• Increase (2001-2011): Added 181.5 million people, a growth rate of 17.7%.
• Rural vs. Urban: 833.5 million (68.84%) lived in rural areas, 377.1 million (31.2%) in urban areas.
• Demographics & Ratios
• Sex Ratio: 940 females per 1000 males (up from 933 in 2001).
• Child Sex Ratio (0-6 years): 919 females per 1000 males.
• Population Density: 382 persons per square kilometer.
• Literacy 
• Overall Literacy: 74.04% (up from 64.83% in 2001).
• Male Literacy: 82.14%.
• Female Literacy: 65.46%.

The correct answer is 30 tonnes.

Key Points

• India's Next Generation Launch Vehicle (NGLV) is being developed by the Indian Space Research Organisation (ISRO) to meet future heavy payload demands for space missions.
• The NGLV is designed with a modular architecture and is capable of carrying up to 30 tonnes of payload to Low Earth Orbit (LEO).
• This capability ensures that the NGLV can support a wide range of space applications, including satellite launches, human spaceflight missions, and interplanetary missions.
• The NGLV is envisioned as a successor to the Polar Satellite Launch Vehicle (PSLV) and Geosynchronous Satellite Launch Vehicle (GSLV) families, offering higher payload capacities and enhanced reliability.
• The development of the NGLV is part of India's long-term strategy to remain competitive in the global space launch market while also addressing domestic requirements.

Additional Information

• Low Earth Orbit (LEO)
• LEO refers to an orbit around Earth with an altitude between 160 km and 2,000 km.
• It is commonly used for Earth observation satellites, communication satellites, and the International Space Station (ISS).
• LEO missions require launch vehicles capable of precise navigation and sufficient payload capacity to meet mission objectives.

The correct answer is It was hosted at the K.D. Jadhav Indoor Hall, IG Stadium, New Delhi, from January 14 to 19, 2025, as part of the BWF Super 750 tier..

Key Points

• The Yonex-Sunrise India Open 2025 is classified as a BWF Super 750 tournament, which is part of the prestigious BWF World Tour.
• The tournament was hosted at the K.D. Jadhav Indoor Hall, located within IG Stadium, New Delhi.
• The event's official schedule spanned from January 14 to 19, 2025, showcasing top international badminton players.
• The BWF Super 750 tier is among the highest levels in the badminton tournament hierarchy, second only to Super 1000 tournaments.
• This tournament attracts global participants and offers significant ranking points and prize money, contributing to its importance in the badminton calendar.

Additional Information

• Yonex-Sunrise India Open
• The Yonex-Sunrise India Open is an annual badminton tournament held in India, featuring international players competing in singles and doubles categories.
• It forms part of the BWF World Tour and is classified under different tiers based on prize money and ranking points.
• The tournament plays a vital role in promoting badminton in India and fostering global competitive standards.

The correct answer is Jharkhand.

Key Points

• The Sindri fertilizer plant is located in Sindri, Jharkhand, which is known for its proximity to the Dhanbad coal belt, one of India's richest coal-producing regions.
• Established in 1944, the Sindri plant is one of India's oldest fertilizer plants and was pivotal in boosting agricultural productivity during its operational years.
• The plant was originally operated by the Fertilizer Corporation of India (FCI) and produced ammonium sulfate and urea, which are essential chemical fertilizers for agriculture.
• Jharkhand's rich coal reserves provided the energy required for the operation of the plant, making Sindri an ideal location for the production of fertilizers.
• The plant underwent modernization and revival under the National Fertilizers Limited (NFL) and Hindustan Urvarak and Rasayan Limited (HURL) initiatives to meet the growing demand for fertilizers in India.

Additional Information

• West Bengal
• West Bengal is known for its industrial hubs and fertile Gangetic plains.
• Important industries in West Bengal include jute, tea, and steel production.
• The state is a significant producer of coal, with the Raniganj coalfield being a major mining area.
• Odisha
• Odisha is rich in mineral resources like iron ore, bauxite.
• The state hosts the Paradip Phosphates Limited (PPL), a major fertilizer company specializing in phosphate-based fertilizers.
• Odisha is also known for its steel plants and aluminum production facilities.
• Bihar
• Bihar is primarily an agricultural state with fertile soil and abundant water resources.
• Bihar's economy relies heavily on agriculture, with crops like wheat, maize, and sugarcane being the primary products.

The correct answer is DNA → RNA → Protein.

Key Points

• The correct sequence of the flow of genetic information is DNA → RNA → Protein, commonly referred to as the Central Dogma of Molecular Biology.
• DNA (Deoxyribonucleic Acid) acts as the genetic material, storing all the information necessary for the synthesis of proteins.
• RNA (Ribonucleic Acid) is synthesized from DNA in a process called transcription. This RNA (typically mRNA) acts as a template for protein synthesis.
• Protein synthesis, also known as translation, occurs at the ribosome where mRNA is decoded to assemble amino acids into a polypeptide chain.
• This process is essential for cellular functions, as proteins serve as enzymes, structural components, and signaling molecules in all living organisms.

Additional Information

• Central Dogma of Molecular Biology
• The concept was first proposed by Francis Crick in 1958.
• It explains the flow of genetic information within a biological system, stating that DNA is transcribed into RNA, and RNA is translated into protein.
• This principle is fundamental to molecular biology and underlies the mechanism of gene expression.

The correct answer is Stamp duties levied by Union but collected by States.

Key Points

• Article 268 of the Indian Constitution primarily deals with duties levied by the Union Government but collected and appropriated by the States.
• It includes stamp duties on certain instruments and excise duties on medicinal and toilet preparations that are levied by the Union but collected by the State.
• The revenue collected from these duties does not form part of the Consolidated Fund of India but is assigned to the States where it is collected.
• This provision ensures that certain revenue sources are shared between the Union and State Governments to balance fiscal responsibilities.
• Examples of stamp duties include duties on transactions such as sale deeds, transfer of property, promissory notes, etc.

Additional Information

• Fiscal Federalism in India:
• The Indian Constitution establishes a clear division of fiscal powers between the Union and State Governments to ensure financial autonomy and accountability.
• Articles 268 to 281 of the Constitution outline the distribution of tax revenues between the Union and the States.
• Article 268 specifically ensures that certain duties levied by the Union are collected and utilized by the States to support their fiscal responsibilities.

The correct answer is Option 1) 0.5 m.

Key Points

• The relationship between the speed of sound, frequency, and wavelength is given by the formula: speed = frequency × wavelength.
• Here, the speed of sound is 340 m/s, and the frequency is 680 Hz.
• To calculate the wavelength, rearrange the formula: wavelength = speed ÷ frequency.
• Substituting the values: wavelength = 340 ÷ 680 = 0.5 m.
• Thus, the wavelength of the sound wave is 0.5 meters, which corresponds to Option 1.

Additional Information

• Speed, Frequency, and Wavelength:
• The speed of a wave is the product of its frequency and wavelength: speed = frequency × wavelength.
• Frequency is the number of wave cycles passing through a point in one second, measured in Hertz (Hz).
• Wavelength is the distance between two consecutive points in phase (e.g., crests or troughs) on a wave, measured in meters.
• This relationship is fundamental in the study of wave mechanics and applies to all types of waves, including sound waves, light waves, and water waves.

The correct answer is Export infra development.

Key Points

• TIES (Trade Infrastructure for Export Scheme) was launched by the Government of India in March 2017 to address the gaps in export infrastructure in the country.
• The scheme aims to create and upgrade export-related infrastructure for promoting exports in India.
• It provides financial assistance to central and state government agencies for the development of necessary export infrastructure, such as testing labs, cold chains, warehouses, and trade promotion centers.
• The funding under TIES is in the form of grants-in-aid, typically not exceeding 50% of the total project cost for eligible projects.
• The scheme focuses on projects that have a direct impact on improving the export competitiveness of goods and services from India.

Additional Information

• Cold storage for dairy (Option 2):
• While cold storage is an essential part of the dairy supply chain, it is not directly related to the TIES scheme.
• Cold storage for dairy products is typically supported under other initiatives like the Pradhan Mantri Kisan SAMPADA Yojana or schemes promoted by the Ministry of Food Processing Industries.
• These initiatives aim to reduce food wastage, improve processing infrastructure, and ensure better returns for farmers.
• PMFBY support (Option 3):
• The Pradhan Mantri Fasal Bima Yojana (PMFBY) is a crop insurance scheme launched in 2016 to provide financial support to farmers in the event of crop failure due to natural calamities.
• It focuses on stabilizing farmer incomes and ensuring the adoption of innovative agricultural practices.
• PMFBY is unrelated to TIES, as it deals with crop insurance rather than export infrastructure.
• Drug pricing regulation (Option 4):
• Drug pricing regulation in India is overseen by the National Pharmaceutical Pricing Authority (NPPA).
• The NPPA ensures the availability of essential medicines at affordable prices and monitors the prices of drugs under the Drug (Prices Control) Order.
• It is not connected to the objectives of the TIES scheme, which is focused on export infrastructure development.

The correct answer is Option 3: A is true, but R is false.

Key Points

• The Happiness Index measures various factors, including political freedom, social support, income, health, life expectancy, and generosity.
• The Happiness Index is officially known as the World Happiness Report, which ranks countries based on these criteria.
• Political freedom and social support are significant components assessed in this index to evaluate the well-being of individuals in a society.
• The World Happiness Report is not published by the UNDP; instead, it is prepared by the Sustainable Development Solutions Network (SDSN) under the auspices of the United Nations.
• The report is based on data collected from global surveys like the Gallup World Poll, emphasizing subjective well-being and happiness levels across countries.

Additional Information

• Key UNDP Indices:
• Human Development Index (HDI): A summary measure of average achievement in:
• A long and healthy life (Life Expectancy at Birth).
• Access to knowledge (Mean Years of Schooling & Expected Years of Schooling).
• A decent standard of living (Gross National Income per Capita).
• Multidimensional Poverty Index (MPI): Identifies overlapping deprivations in health, education, and living standards (like nutrition, sanitation, electricity, cooking fuel, etc.) for individuals.
• Gender Inequality Index (GII): Measures gender disparities in reproductive health, empowerment, and economic activity.
• Inequality-adjusted Human Development Index (IHDI): Accounts for inequality within countries by "discounting" the HDI in proportion to the level of inequality, notes United Nations Development Programme.
• Gender Development Index (GDI): Compares HDI for men and women.
• Planetary Pressures-adjusted HDI (PHDI): Adjusts the HDI for environmental impact.
• Gender Social Norms Index (GSNI): Measures social norms restricting gender equality.

The correct answer is Store CO₂ in vegetation and soil.

Key Points

• Carbon sink enhancement refers to increasing the capacity of ecosystems like forests, grasslands, and soil to absorb and store carbon dioxide (CO₂) from the atmosphere.
• India’s climate plan aims to achieve an additional 2.5–3 billion tonnes of CO₂ equivalent sequestration through afforestation and other green initiatives by 2030.
• This approach supports India’s commitment under the Paris Agreement to mitigate greenhouse gas emissions and combat climate change.
• Carbon sinks help reduce the concentration of CO₂ in the atmosphere, thereby minimizing the harmful effects of global warming.
• Vegetation and soil act as natural reservoirs, storing carbon for extended periods through processes like photosynthesis and organic matter decomposition.

Additional Information

• Deforestation
• Deforestation is the process of clearing forests for agriculture, urbanization, or industrial use.
• It leads to increased CO₂ emissions as trees, which act as carbon sinks, are removed.
• Burn more fossil fuels
• Burning fossil fuels like coal, oil, and gas releases significant amounts of CO₂ and other greenhouse gases into the atmosphere.
• It exacerbates global warming and is the opposite of carbon sink enhancement.
• Monocultures
• Monocultures involve planting a single crop species over a large area, which can lead to soil degradation and reduced biodiversity.
• Although monocultures may absorb CO₂, they are less effective as carbon sinks compared to diverse ecosystems like forests.

The correct answer is Option 1.

Key Points

• Import substitution refers to an economic policy aimed at reducing dependency on foreign goods by promoting domestic production and industries.
• India adopted the policy of import substitution during the planning era, particularly during the Second Five-Year Plan (1956-61) led by P.C. Mahalanobis, to boost industrialization.
• The primary objective was to protect the domestic industries, which were in their nascent stages, from foreign competition and to achieve self-reliance in industrial production.
• Through high import tariffs, quotas, and trade restrictions, foreign goods were made less competitive, encouraging the growth of Indian industries.
• While this policy played a critical role in laying the foundation for India’s industrial base, it also led to inefficiencies and a lack of competition in the long run.

Additional Information

• First FYP (1951-56): Focused on agriculture, controlling inflation, and refugee rehabilitation (Harrod-Domar model).
• Second FYP (1956-61): Emphasized rapid industrialization, especially heavy industries (Mahalanobis Model).
• Third FYP (1961-66): Aimed for food self-sufficiency and increased agricultural/industrial production, but faced challenges.
• Fifth FYP (1974-78): Key focus on Garibi Hatao (poverty alleviation), employment, and social justice.
• Sixth FYP (1980-85): Introduced early economic liberalization and aimed for technological self-reliance.
• Eighth FYP (1992-97): Launched after the 1991 economic crisis, promoting LPG (Liberalisation, Privatisation, Globalisation) and industrial modernization.
• Ninth FYP (1997-2002): Focused on social sector development and equitable growth.
• Tenth FYP (2002-07): Aimed for faster and more inclusive growth.
• Eleventh FYP (2007-12): Focused on faster, more inclusive, and sustainable growth.
• Twelfth FYP (2012-17): The final FYP, targeting faster, sustainable, and more inclusive growth with emphasis on education, infrastructure, and poverty reduction.

The correct answer is Option 2: Statement 2 only.

Key Points

• Pulsars are highly magnetized, rotating neutron stars that emit beams of electromagnetic radiation from their poles.
• The radiation emitted by pulsars is observable as pulses due to their rotation, hence the name "pulsars."
• Pulsars were first discovered in 1967 by Jocelyn Bell Burnell and Antony Hewish and are considered some of the most precise natural clocks in the universe.
• Pulsars are formed during the supernova explosion of massive stars, resulting in the collapse of their cores into dense neutron stars.
• These celestial objects rotate rapidly, with some pulsars completing hundreds of rotations per second (millisecond pulsars).

Additional Information

• Black Holes:
• Black holes are regions of spacetime exhibiting gravitational forces so strong that nothing, not even light, can escape from them.
• The event horizon is the boundary around a black hole beyond which nothing can return.
• They are classified into stellar-mass black holes, supermassive black holes, and intermediate black holes based on their size and formation.
• Pulsars:
• Pulsars are a subtype of neutron stars and are among the densest objects in the universe.
• The precise timing of pulsar emissions allows scientists to study gravitational waves, test theories of relativity, and detect exoplanets.
• Some pulsars are part of binary systems, interacting with companion stars and exhibiting phenomena such as X-ray pulsations.

The correct answer is Article 250.

Key Points

• Article 250 of the Indian Constitution empowers the Parliament to legislate on matters enumerated in the State List under exceptional circumstances, such as during a National Emergency.
• While a National Emergency is in operation (as per Article 352), Parliament can make laws on subjects in the State List for the entire country or part of its territory.
• Such laws made by Parliament under Article 250 remain valid only during the period of the emergency and up to six months after it ceases to operate.
• This provision ensures that Parliament can address urgent national concerns that might require overriding state authority during an emergency.
• Article 250 is a crucial component of the emergency provisions in the Indian Constitution, allowing the central government to maintain legislative control during extraordinary situations.

Additional Information

• Article 256
• Article 256 deals with the obligation of states and the Union to ensure compliance with laws made by Parliament.
• It mandates that the executive power of every state shall be exercised to ensure compliance with the laws made by Parliament.
• It does not specifically address the legislative powers of the Parliament during an emergency.
• Article 254
• Article 254 addresses inconsistencies between laws made by Parliament and laws made by state legislatures.
• If there is a conflict between a central law and a state law on a subject in the Concurrent List, the central law prevails.
• It is unrelated to Parliament’s ability to legislate on State List subjects during a National Emergency.
• Article 352
• Article 352 provides for the proclamation of a National Emergency in India.
• The President can declare a National Emergency if there is a threat to the security of India or any part of it due to war, external aggression, or armed rebellion.
• While Article 352 enables the declaration of an emergency, it is Article 250 that specifically allows Parliament to legislate on State List subjects during such emergencies.

The correct answer is Option 2.

Key Points

• Article 131A was introduced by the 42nd Constitutional Amendment Act of 1976 during the Emergency period in India.
• This article granted the Supreme Court exclusive jurisdiction to decide the constitutional validity of any Central law, bypassing High Courts.
• The provision aimed to centralize judicial power in the Supreme Court, reducing the role of High Courts in this matter.
• The 43rd Constitutional Amendment Act of 1977 repealed Article 131A, restoring the original balance of judicial power between the High Courts and the Supreme Court.
• Thus, Statement 2 is correct, while Statement 1 is incorrect.

Additional Information

• 42nd Constitutional Amendment Act, 1976:
• Known as the "Mini-Constitution," this amendment introduced sweeping changes to the Constitution, increasing the power of the Central government.
• It added Article 131A, which centralized the adjudication of the validity of Central laws in the Supreme Court.
• Other changes included the addition of the Fundamental Duties (Part IVA) and the extension of the term of Lok Sabha and State Assemblies from 5 to 6 years.
• 43rd Constitutional Amendment Act, 1977:
• This amendment reversed many provisions of the 42nd Amendment, reflecting the shift in political power after the Emergency.
• It repealed Article 131A, restoring the High Courts' jurisdiction in deciding the validity of Central laws.
• The amendment also reinstated judicial review and reduced the power of the executive in certain areas.
• Judicial Review:
• Judicial review is the power of the judiciary to examine the constitutionality of legislative and executive actions.
• In India, both the Supreme Court and High Courts have the power of judicial review under Articles 32 and 226 of the Constitution.
• It ensures the supremacy of the Constitution and protects fundamental rights.

Shortcut Trick

Break the given decimal into its integer and fractional parts:

3.375 = 3 + 0.375

Recall the standard decimal conversion for eighths: 0.375 = 3/8

⇒ 3.375 = 3 + 3/8 = (3 × 8 + 3) ÷ 8

⇒ 3.375 = 27/8

∴ The correct answer is 27/8.

Alternate Method

Given:

Decimal value = 3.375

Method Used:

To convert a decimal to a fraction, write the entire number without the decimal point over 1 followed by as many zeros as there are decimal places, and then simplify.

Calculations:

⇒ Since 3.375 has 3 decimal places, we write it over 1000.

⇒ 3.375 = 3375 / 1000

⇒ To simplify, divide both the numerator and denominator by 25:

⇒ Numerator: 3375 ÷ 25 = 135

⇒ Denominator: 1000 ÷ 25 = 40

⇒ The fraction becomes 135 / 40.

⇒ Now, divide both by 5 to reduce it further:

⇒ Numerator: 135 ÷ 5 = 27

⇒ Denominator: 40 ÷ 5 = 8

⇒ The fraction in its simplest form is 27/8.

∴ The correct answer is 27/8.

Additional Information

Common Decimal Equivalents

Memorizing common base-8 fractions speeds up calculations in exams: 1/8 = 0.125, 3/8 = 0.375, 5/8 = 0.625, and 7/8 = 0.875.

Proper vs Improper Fractions

A proper fraction has a numerator smaller than its denominator (e.g., 3/8). An improper fraction has a numerator greater than or equal to its denominator (e.g., 27/8). Mixed numbers can be formed directly from improper fractions.

Converting Repeating Decimals

For non-terminating but repeating decimals, place the repeating digits over an equal number of 9s instead of 10s. Example: 0.444... = 4/9, and 0.3636... = 36/99 = 4/11.

Shortcut Trick

Given, 15% of a = 25% of b

Directly equate the percentage values: 15 × a = 25 × b

Rearranging the terms, we get the ratio: a ÷ b = 25 ÷ 15

Simplifying the fraction gives: a : b = 5 : 3

∴ The correct answer is 5 : 3.

Alternate Method

Given:

15% of a = 25% of b

Visualising the Ratio

Value of a (5 Parts)

Value of b (3 Parts)

Ratio a : b = 5 : 3

Calculations:

⇒ (15 ÷ 100) × a = (25 ÷ 100) × b

⇒ 15 × a = 25 × b

⇒ a ÷ b = 25 ÷ 15

⇒ a ÷ b = 5 ÷ 3

⇒ a : b = 5 : 3

∴ The correct answer is 5 : 3.

Additional Information

Ratio and Proportion Basics

A ratio indicates how many times one number contains another. The ratio of x to y is mathematically represented as x : y or x ÷ y.

Inverse Relationship

If p × A = q × B, then the ratio of A to B is simply q : p. This happens because crossing over the terms yields A ÷ B = q ÷ p.

Percentage to Fraction

To convert a percentage into a fraction, divide the percentage value by 100. For example, 25% is equivalent to 25 ÷ 100, which simplifies to 1 ÷ 4.

Shortcut Trick

To find the number x to be subtracted from a, b, c, and d to make them proportional, use the direct formula:

x = (ad − bc) ÷ ((a + d) − (b + c))

Here, a = 7, b = 9, c = 8, d = 11.

x = (7 × 11 − 9 × 8) ÷ ((7 + 11) − (9 + 8))

x = (77 − 72) ÷ (18 − 17) = 5 ÷ 1 = 5

∴ The correct answer is 5.

Alternate Method

Given:

The numbers are 7, 9, 8, 11.

Formula Used:

If four numbers A, B, C, D are in proportion, then A ÷ B = C ÷ D.

Cross Multiplication for Proportion

7 − x

9 − x

8 − x

11 − x

=

(7 − x)(11 − x) = (9 − x)(8 − x)

Calculations:

Let the number to be subtracted be x.

⇒ (7 − x) ÷ (9 − x) = (8 − x) ÷ (11 − x)

⇒ (7 − x)(11 − x) = (9 − x)(8 − x)

⇒ 77 − 11x − 7x + x² = 72 − 9x − 8x + x²

⇒ 77 − 18x = 72 − 17x

⇒ 77 − 72 = 18x − 17x

⇒ x = 5

∴ The correct answer is 5.

Additional Information

Continued Proportion

If three numbers a, b, c are in continued proportion, then a ÷ b = b ÷ c, which means b² = ac. Here, b is known as the mean proportional.

Componendo and Dividendo

If a ÷ b = c ÷ d, then according to this rule: (a + b) ÷ (a − b) = (c + d) ÷ (c − d). This is a highly useful tool for simplifying complex ratio equations.

Fourth Proportional

For any three given numbers a, b, and c, their fourth proportional is a number d such that a ÷ b = c ÷ d, giving the result d = (b × c) ÷ a.

Shortcut Trick

Convert the fractional part to a decimal directly: 5/8 = 0.625.

So, 2 ⁵/₈ = 2.625.

Given expression: 6.4 − (2.625 + 1.75)

⇒ 6.4 − 4.375 = 2.025

∴ The correct answer is 2.025.

Alternate Method

Given:

6.4 − (2 ⁵/₈ + 1.75)

Concept Used:

Follow the BODMAS rule to solve the expression in the brackets first.

Calculations:

⇒ Convert 2 ⁵/₈ to decimal: (2 × 8 + 5) ÷ 8 = 21 ÷ 8 = 2.625

⇒ Substitute the decimal value into the given expression:

⇒ 6.4 − (2.625 + 1.75)

⇒ Solve the addition inside the bracket:

⇒ 2.625 + 1.75 = 4.375

⇒ Subtract the calculated sum from 6.4:

⇒ 6.400 − 4.375 = 2.025

∴ The correct answer is 2.025.

Additional Information

BODMAS Rule

It determines the correct sequence of operations in any mathematical expression: Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Common Fraction to Decimal Conversions

Memorizing eighth-based fractions saves time in calculations: 1/8 = 0.125, 3/8 = 0.375, 5/8 = 0.625, and 7/8 = 0.875.

Decimal Addition & Subtraction

Always align the decimal points vertically before calculating. You can append trailing zeros to numbers (e.g., writing 6.4 as 6.400) to balance the number of decimal places without changing the value.

Shortcut Trick

Profit is distributed in the ratio of (Capital × Time).

Alok's total capital = (90,000 × 6) + (1,20,000 × 6) = 12,60,000

Deepak's time in business = 12 − 3 (joined late) − 3 (left early) = 6 months.

Deepak's total capital = 1,20,000 × 6 = 7,20,000

Ratio = 12,60,000 : 7,20,000 = 7 : 4

∴ The correct answer is 7:4.

Alternate Method

0m 3m 6m 9m 12m Alok 90k × 6m 120k × 6m Deepak 120k × 6m

Given:

Alok's initial investment = ₹90,000

Alok's increased investment after 6 months = ₹90,000 + ₹30,000 = ₹1,20,000

Deepak's investment = ₹1,20,000

Total time for business = 12 months

Formula Used:

Profit Ratio = (C₁ × T₁) : (C₂ × T₂)

Calculations:

⇒ Alok invested ₹90,000 for the first 6 months.

⇒ Alok invested ₹1,20,000 for the remaining 6 months.

⇒ Equivalent capital for Alok = (90,000 × 6) + (1,20,000 × 6)

⇒ Equivalent capital for Alok = 5,40,000 + 7,20,000 = 12,60,000

⇒ Deepak joined after 3 months and left 3 months before the end.

⇒ Deepak's total investment time = 12 − 3 − 3 = 6 months.

⇒ Equivalent capital for Deepak = 1,20,000 × 6 = 7,20,000

⇒ Profit sharing ratio (Alok : Deepak) = 12,60,000 : 7,20,000

⇒ Alok : Deepak = 126 : 72 = 7 : 4

∴ The correct answer is 7:4.

Additional Information

Joining Mid-way

If a partner joins the business after t months, their investment is considered active for (12 − t) months in an annual profit cycle.

Compound Ratio in Partnership

The profit is always distributed in the compound ratio of individual investments and time periods: (C₁ × T₁) : (C₂ × T₂) : (C₃ × T₃).

Active and Sleeping Partner

An active partner is paid a fixed amount or a specific percentage of the profit for managing the operations, and the remaining profit is distributed based on their capital and time ratios.

Shortcut Trick

Teacher's Age = New Average + (Old Members × Increase in Average)

Old Members = 19 students

Increase in Average = 18 − 17 = 1 year

Teacher's Age = 18 + (19 × 1) = 37 years

∴ The correct answer is 37 years.

Alternate Method

Given:

Average age of 19 students = 17 years

New average age after teacher joins = 18 years

Formula Used:

Sum of values = Average × Number of values

323 years

19 students (19 × 17)

19 students Teacher

20 members total = 20 × 18 = 360 years

360 − 323 = 37

Calculations:

⇒ Total age of 19 students = 19 × 17 = 323 years

⇒ Number of members after teacher joins = 19 + 1 = 20 members

⇒ Total age of 20 members = 20 × 18 = 360 years

⇒ Teacher's age = Total age of 20 members − Total age of 19 students

⇒ Teacher's age = 360 − 323 = 37 years

∴ The correct answer is 37 years.

Additional Information

Combined Average

When two groups of sizes n₁ and n₂ with averages A₁ and A₂ are mixed, Combined Average = (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Exclusion of a Member

If a person leaves the group, Age of leaving person = Old Average − (New Total Members × Change in Average). Keep the positive or negative sign of the change intact.

Replacement of a Member

When a new person replaces an old one, Age of new person = Age of removed person + (Total Members × Increase in Average).

Shortcut Trick

Deviation of 30 employees from overall average = ₹72,000 − ₹62,000 = +₹10,000 each.

Total extra deviation for these 30 employees = 30 × 10,000 = +₹3,00,000.

To maintain the overall average, the remaining 50 employees must compensate by −₹3,00,000.

Average deviation for remaining employees = −₹3,00,000 ÷ 50 = −₹6,000.

Required average = ₹62,000 − ₹6,000 = ₹56,000.

∴ The correct answer is ₹56,000.

Alternate Method

Given:

Total number of employees = 80

Overall average salary = ₹62,000

Number of employees in the first group = 30

Average salary of the first group = ₹72,000

Total: 80 employees | Avg = ₹62,000

30 employees Avg: ₹72,000

50 employees Avg: ?

Formula Used:

Total Sum = Average × Number of Observations

Average of Remaining = Sum of Remaining ÷ Number of Remaining Observations

Calculations:

⇒ Total salary of 80 employees = 80 × 62000 = ₹49,60,000

⇒ Total salary of 30 employees = 30 × 72000 = ₹21,60,000

⇒ Remaining employees = 80 − 30 = 50

⇒ Total salary of remaining 50 employees = 49,60,000 − 21,60,000 = ₹28,00,000

⇒ Average salary of remaining employees = 28,00,000 ÷ 50 = ₹56,000

∴ The correct answer is ₹56,000.

Additional Information

Combined Average

When two groups of sizes n₁ and n₂ have averages A₁ and A₂, the combined average is given by: A = (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Net Deviation Method

When checking against a mean average, the sum of deviations of all elements from that mean is always zero. This principle is extremely helpful for calculating missing values quickly without doing large multiplications.

Weighted Average Trick

Instead of using raw numbers, you can use the ratio of the group sizes (e.g., 30 : 50 = 3 : 5) in your formulas to further simplify and speed up calculation steps.

Shortcut Trick

In Compound Interest, if a sum becomes n times in t years, it becomes n^x times in xt years.

The sum becomes 4 times in 6 years.

We need to find the time taken for it to become 16 times.

Since 16 = 4² times, we multiply the time by 2.

Required time = 6 × 2 = 12 years.

∴ The correct answer is 12 years.

Alternate Method

P 4P 16P

6 years

6 years

× 4

× 4

Total: 12 years

Given:

A principal sum P becomes 4P in 6 years at Compound Interest.

Formula Used:

A = P(1 + R/100)^T

Where A is the Amount, P is the Principal, R is the Rate of interest, and T is the Time in years.

Calculations:

Let the original sum be P.

According to the given condition, the amount becomes 4 times in 6 years:

⇒ 4P = P(1 + R/100)⁶

⇒ 4 = (1 + R/100)⁶

We need to find the time T when the amount becomes 16P:

⇒ 16P = P(1 + R/100)^T

⇒ 16 = (1 + R/100)^T

We can write 16 as 4²:

⇒ 4² = (1 + R/100)^T

Substitute the value of 4 from the first equation:

⇒ ((1 + R/100)⁶)² = (1 + R/100)^T

⇒ (1 + R/100)¹² = (1 + R/100)^T

By equating the powers on both sides, we get:

⇒ T = 12 years

∴ The correct answer is 12 years.

Additional Information

Compound Interest Multiplier Concept

In Compound Interest, the amount grows geometrically over equal periods. If a sum becomes n times in t years, it will become n² times in 2t years, n³ times in 3t years, and so on.

Half-Yearly Compounding

When interest is compounded half-yearly, the annual rate is halved (R/2) and the total number of periods is doubled (2T) in the CI formula.

Difference Between SI and CI

The difference between SI and CI on a given sum P at a rate R% for 2 years is directly calculated as: Difference = P(R/100)².

Shortcut Trick

Given the ratio of Loss to Cost Price (CP) = 1 : 10.

Let the Loss be 1 unit and the Cost Price be 10 units.

Selling Price (SP) = Cost Price − Loss = 10 − 1 = 9 units.

The ratio of Selling Price to Cost Price = SP : CP = 9 : 10.

∴ The correct answer is 9:10.

Alternate Method

Given:

Ratio of Loss to Cost Price (CP) = 1 : 10

Formula Used:

Selling Price (SP) = Cost Price (CP) − Loss

SP = 9x Loss = x

Cost Price (CP) = 10x

Ratio of SP to CP = 9 : 10

Calculations:

⇒ Let the loss incurred be x and the Cost Price (CP) be 10x.

⇒ SP = CP − Loss

⇒ SP = 10x − x = 9x

⇒ Required Ratio = SP : CP

⇒ SP : CP = 9x : 10x

⇒ SP : CP = 9 : 10

∴ The correct answer is 9:10.

Additional Information

Cost Price and Selling Price

Cost Price (CP) is the price at which the item is purchased by a seller, and Selling Price (SP) is the price at which it is sold to the customer.

Profit and Loss Logic

When Selling Price > Cost Price, there is a Profit calculated as (SP − CP). When Cost Price > Selling Price, there is a Loss calculated as (CP − SP).

Percentage Calculation

Both profit and loss are always evaluated against the base Cost Price (CP). Profit % = (Profit ÷ CP) × 100 and Loss % = (Loss ÷ CP) × 100.

Shortcut Trick

The ratio of Cost Price (CP) to Marked Price (MP) can be directly calculated by the formula:

CP / MP = (100 − Discount%) / (100 + Profit%)

⇒ CP / 150 = (100 − 10) / (100 + 25)

⇒ CP / 150 = 90 / 125 = 18 / 25

⇒ CP = (18 / 25) × 150 = 18 × 6 = ₹108

∴ The correct answer is ₹108.

Alternate Method

Given:

Marked Price (MP) = ₹150, Discount = 10%, Profit = 25%

Formula Used:

Selling Price (SP) = MP × (100 − Discount%) ÷ 100

Cost Price (CP) = SP × 100 ÷ (100 + Profit%)

CP ₹108

SP ₹135

MP ₹150

+25% Profit

-10% Discount

Cost Price = ₹108

Calculations:

⇒ SP = 150 × (100 − 10) ÷ 100

⇒ SP = 150 × 90 ÷ 100 = ₹135

⇒ CP = 135 × 100 ÷ (100 + 25)

⇒ CP = 135 × 100 ÷ 125

⇒ CP = 135 × 4 ÷ 5

⇒ CP = 27 × 4 = ₹108

∴ The correct answer is ₹108.

Additional Information

Marked Price and Discount

Discount is always calculated on the Marked Price (MP). The resulting value is the Selling Price: SP = MP − Discount.

Profit and Loss Calculation

Profit or Loss is always calculated on the Cost Price (CP). Thus, SP = CP × (100 + Profit%) ÷ 100.

Successive Discounts

If two successive discounts of d₁% and d₂% are given, the equivalent single discount is (d₁ + d₂ − d₁×d₂÷100)%.

Shortcut Trick

Let the Marked Price (MP) be 100 units.

Discounted price = 100 − 20% = 80 units.

Cost Price (CP) including fee = 80 + (15% of 80) = 92 units.

Profit = 25% of 92 units = 23 units.

Given 23 units = ₹7800 ⇒ 100 units = (7800 ÷ 23) × 100 = ₹33,913.04.

∴ The correct answer is ₹33,913.04.

Alternate Method

Given:

Discount = 20%

Installation and support fee = 15%

Profit made by reseller = 25%

Amount of profit = ₹7800

MP (100x)−20%80x+15%CP (92x)+25%SPProfit = 23x = ₹7800

Calculation:

⇒ Let the Marked Price (MP) of the license = 100x

⇒ Price to the reseller after 20% discount = 100x − (20% of 100x) = 80x

⇒ Final Cost Price (CP) for the reseller = 80x + (15% of 80x) = 80x + 12x = 92x

⇒ Profit earned by the reseller = 25% of CP = 25% of 92x = 23x

⇒ Since the profit earned is ₹7800

⇒ 23x = 7800

⇒ x = 7800 ÷ 23

⇒ Original MP = 100x = 100 × (7800 ÷ 23) = 780000 ÷ 23

⇒ Original MP = 33913.043...

∴ The correct answer is ₹33,913.04.

Additional Information

Marked Price and Discount

Discount is always calculated on the Marked Price (MP). Selling Price (SP) = MP × (100 − Discount%) ÷ 100.

Profit and Loss

Profit or loss is always calculated on the Cost Price (CP) unless stated otherwise. Profit = SP − CP.

Effective Cost Price

Any additional overheads like transportation, installation fees, or repairs are added to the initial purchase value to determine the final effective Cost Price.

Shortcut Trick

Initial Milk = (7/9) × 90 = 70 L, Water = (2/9) × 90 = 20 L.

Since the same amount 'x' is added to both, the difference between their quantities remains unchanged.

Initial difference = 70 − 20 = 50 L.

New ratio is 5 : 2, so the difference is 5 − 2 = 3 parts, which means 3 parts = 50 L.

Water added (x) = New water (2 parts) − Old water = (100/3) − 20 = 40/3 = 13.33 L.

∴ The correct answer is 13.33 litres.

Alternate Method

Given:

Total mixture = 90 litres

Initial ratio of Milk to Water = 7 : 2

New ratio of Milk to Water after adding 'x' litres of both = 5 : 2

Formula Used:

Ratio = Quantity of Component 1 ÷ Quantity of Component 2

Initial (7:2) Final (5:2)

Milk: 70L

W: 20L

Milk: (70 + x)L

W: (20 + x)L

Adding 'x' litres of both changes ratio to 5 : 2

Calculations:

⇒ Quantity of milk initially = (7/9) × 90 = 70 litres

⇒ Quantity of water initially = (2/9) × 90 = 20 litres

⇒ After adding 'x' litres of each, new Milk = (70 + x) and new Water = (20 + x)

⇒ According to the given condition:

⇒ (70 + x) / (20 + x) = 5 / 2

⇒ Cross-multiplying: 2 × (70 + x) = 5 × (20 + x)

⇒ 140 + 2x = 100 + 5x

⇒ 5x − 2x = 140 − 100

⇒ 3x = 40

⇒ x = 40 / 3 = 13.33 litres

∴ The correct answer is 13.33 litres.

Additional Information

Constant Difference Method

Whenever equal amounts of two components are added or removed from a mixture, the absolute difference between their quantities remains strictly constant. This acts as a powerful shortcut by equating the part difference of the ratios.

Cross-Multiplication Principle

For equations of the form (a + x) / (b + x) = c / d, the difference between the terms translates directly to: x(c − d) = ad − bc, allowing faster mental calculation without writing intermediate steps.

Mixtures & Replacement Concept

If 'y' litres of a mixture is removed and replaced by 'y' litres of one of the components (e.g., water), the ratio of the components in the remaining mixture right before replacement stays exactly identical to the initial original ratio.

Shortcut Trick

The volume of a cylinder that completely circumscribes a sphere is exactly (3/2) times the volume of the sphere.

Ratio of Cylinder Volume to Sphere Volume = 3 : 2.

Vacant Space = Cylinder Volume − Sphere Volume = (3/2 − 1) = 1/2 of Sphere Volume.

Vacant Space = (1/2) × 36π = 18π cm³.

∴ The correct answer is 18π cm³.

Alternate Method

Given:

Volume of the sphere = 36π cm³

Height and diameter of the cylinder = Diameter of the sphere

Formula Used:

Volume of Sphere = (4/3)πr³

Volume of Cylinder = πr²h

r = 3h = 2r = 6d = 2r = 6

Calculations:

⇒ (4/3)πr³ = 36π

⇒ r³ = 36 × 3 ÷ 4

⇒ r³ = 27

⇒ Radius of sphere (r) = 3 cm

⇒ Radius of cylinder (R) = 3 cm

⇒ Height of cylinder (h) = diameter of sphere = 2 × 3 = 6 cm

⇒ Volume of cylinder = πR²h = π × 3² × 6 = 54π cm³

⇒ Volume of vacant space = Volume of cylinder − Volume of sphere

⇒ Vacant space = 54π − 36π = 18π cm³

∴ The correct answer is 18π cm³.

Additional Information

Ratio of Volumes (Cone, Sphere, Cylinder)

If a cone, a sphere, and a cylinder have the same radius (r) and the same height (2r), their volumes will always be exactly in the ratio of 1 : 2 : 3.

Archimedes' Hat-Box Theorem

A sphere inscribed perfectly inside a right circular cylinder has exactly 2/3 the volume and 2/3 the surface area of the cylinder.

Key Mensuration Formulas

Curved Surface Area of Cylinder = 2πrh | Surface Area of Sphere = 4πr² | Volume of Cone = (1/3)πr²h

Shortcut Trick

When a solid is melted and recast into another shape, their volumes remain equal.

Volume of Hemisphere = Volume of Cone

⇒ ²/₃ × π × R³ = ¹/₃ × π × R² × h

⇒ 2 × R = h

∴ The correct answer is 2R.

Alternate Method

Given:

Radius of the hemisphere = R

Radius of the recast cone = R

Formula Used:

Volume of a hemisphere = ²/₃ × π × r³

Volume of a cone = ¹/₃ × π × r² × h

R Hemisphere

Melt & Recast

R

h Cone

Calculations:

⇒ Let the height of the newly formed cone be h.

⇒ Since the hemisphere is simply melted to form the cone, the original material volume completely transforms into the new volume.

⇒ Volume of Hemisphere = Volume of Cone

⇒ ²/₃ × π × R³ = ¹/₃ × π × R² × h

⇒ Canceling the common terms (¹/₃ and π) from both sides:

⇒ 2 × R³ = R² × h

⇒ Dividing both sides by R²:

⇒ h = 2R

∴ The correct answer is 2R.

Additional Information

Melting and Recasting Property

When any 3D solid is melted and recast into another shape, the total volume of the material remains strictly constant, even though dimensions and total surface area typically change.

Volume Relation of Cylinder & Cone

For a given identical radius (r) and height (h), the volume of a cylinder (V = πr²h) is exactly 3 times the volume of a cone (V = ¹/₃πr²h).

Hemisphere Surface Areas

The Curved Surface Area (CSA) of a solid hemisphere is 2πr², while its Total Surface Area (TSA) includes the flat circular base, resulting in 3πr².

Shortcut Trick

When 3D shapes are melted and recast, their total volume is conserved.

For hemispheres, the volume constant (2/3)π cancels out, giving: R³ = r₁³ + r₂³

R³ = 6³ + 8³ = 216 + 512 = 728

Since 9³ = 729, the approximate value of R is 9.

∴ The correct answer is 9 cm.

Alternate Method

Given:

Radius of first hemisphere (r₁) = 6 cm

Radius of second hemisphere (r₂) = 8 cm

Formula Used:

Volume of a hemisphere = (2/3)πr³

r=6

r=8

=

R=?

V1 + V2 = Vnew

Calculations:

Let the radius of the newly formed hemisphere be R.

⇒ Total volume of melted hemispheres = Volume of the new single hemisphere

⇒ (2/3)π(6)³ + (2/3)π(8)³ = (2/3)πR³

⇒ (2/3)π × (6³ + 8³) = (2/3)πR³

Canceling out (2/3)π from both sides:

⇒ 6³ + 8³ = R³

⇒ 216 + 512 = R³

⇒ R³ = 728

⇒ R = (728)^1/3

⇒ We know that 9³ = 729. Therefore, (728)^1/3 ≈ 9 cm.

∴ The correct answer is 9 cm.

Additional Information

Volume of a Sphere

The volume of a full sphere is (4/3)πr³, which is exactly double the volume of a hemisphere.

Surface Area of a Hemisphere

The Curved Surface Area (CSA) of a hemisphere is 2πr². The Total Surface Area (TSA) includes the flat circular base and is 3πr².

Concept of Melting and Recasting

When any 3D object is melted and recast into another shape, the Total Volume remains conserved. The sum of the volumes of the smaller parts directly equals the volume of the newly formed object.

Shortcut Trick

For a regular square pyramid, Vertical Height (h) = √(l² − (a/2)²).

⇒ h = √(10² − 6²) = √(100 − 36) = √64 = 8 cm.

Volume = 1/3 × Base Area × h

⇒ Volume = 1/3 × (12 × 12) × 8 = 48 × 8 = 384 cm³.

∴ The correct answer is 384 cm³.

Alternate Method

Given:

⇒ Side of the square base (a) = 12 cm

⇒ Slant height (l) = 10 cm

Formula Used:

⇒ Volume of a right pyramid = 1/3 × Base Area × Vertical Height

⇒ Vertical Height (h) = √(l² − r²) where r is half of the base side length.

Calculations:

⇒ Base Area = a × a = 12 × 12 = 144 cm²

⇒ Half of the base side length (r) = 12 ÷ 2 = 6 cm

⇒ Applying Pythagoras theorem to find the vertical height:

⇒ h² + r² = l²

⇒ h² + 6² = 10²

⇒ h² + 36 = 100

⇒ h² = 64

⇒ h = 8 cm

⇒ Volume = 1/3 × 144 × 8

⇒ Volume = 48 × 8 = 384 cm³

∴ The correct answer is 384 cm³.

Additional Information

Lateral Surface Area of Pyramid

It is the total area of all the triangular lateral faces. Formula: LSA = 1/2 × Perimeter of Base × Slant Height.

Total Surface Area of Pyramid

It comprises both the lateral faces and the base. Formula: TSA = Lateral Surface Area + Base Area.

Slant Height vs Vertical Height

Vertical height is the perpendicular distance from the apex straight down to the center of the base, whereas slant height is measured along the lateral face from the apex to the midpoint of the base edge.

Shortcut Trick

The ratio of the areas of two circles is equal to the square of the ratio of their radii.

Ratio of areas = 1² : (√2)² = 1 : 2

Here, 1 unit → 100 cm²

Larger tyre area = 2 units → 2 × 100 = 200 cm²

∴ The correct answer is 200 cm².

Alternate Method

r Area = 100 cm²

r√2 Area = ?

Given:

Ratio of radii of two tyres = 1 : √2

Area of the smaller tyre (A₁) = 100 cm²

Formula Used:

Area of a circle = πr²

Calculation:

⇒ Let the radii of the smaller and larger tyres be r and r√2 respectively.

⇒ Area of smaller tyre (A₁) = π × (r)² = πr²

⇒ πr² = 100 cm²

⇒ Area of larger tyre (A₂) = π × (r√2)²

⇒ A₂ = π × 2r² = 2πr²

⇒ Substitute the value of πr² into the equation for A₂:

⇒ A₂ = 2 × 100

⇒ A₂ = 200 cm²

∴ The correct answer is 200 cm².

Additional Information

Circumference of Circle

The circumference of a circle is directly proportional to its radius (C = 2πr). The ratio of circumferences of two circles is the same as the ratio of their radii.

Volume of Spheres

For 3D objects like spheres, the volume is proportional to the cube of the radius (V = 4 ÷ 3 × πr³). The ratio of volumes is r₁³ : r₂³.

Area of a Sector

The area of a sector with a central angle θ in a circle of radius r is given by (θ ÷ 360) × πr².

Shortcut Trick

For any triangle, the angle subtended by the incenter at any two vertices is 90° plus half of the third angle.

⇒ ∠BIC = 90° + (∠BAC / 2)

⇒ 125° = 90° + (∠BAC / 2)

⇒ ∠BAC / 2 = 125° − 90°

⇒ ∠BAC = 35° × 2 = 70°

∴ The correct answer is 70°.

Alternate Method

Given:

The angle bisectors of ∠ABC and ∠ACB meet at the incenter I.

Measure of ∠BIC = 125°

Formula Used:

Sum of interior angles of a triangle = 180°

An angle bisector divides an angle into two equal parts.

A B C I 125° ? (70°)

Calculations:

In ΔBIC, the sum of angles is 180°.

⇒ ∠IBC + ∠ICB + ∠BIC = 180°

Since BI and CI are the angle bisectors of ∠ABC and ∠ACB respectively:

⇒ ∠IBC = ∠ABC / 2

⇒ ∠ICB = ∠ACB / 2

Substituting these into the equation for ΔBIC:

⇒ (∠ABC / 2) + (∠ACB / 2) + 125° = 180°

⇒ (∠ABC + ∠ACB) / 2 = 180° − 125°

⇒ (∠ABC + ∠ACB) / 2 = 55°

⇒ ∠ABC + ∠ACB = 55° × 2 = 110°

Now, in ΔABC, the sum of all interior angles is 180°.

⇒ ∠BAC + ∠ABC + ∠ACB = 180°

⇒ ∠BAC + 110° = 180°

⇒ ∠BAC = 180° − 110°

⇒ ∠BAC = 70°

∴ The correct answer is 70°.

Additional Information

Incenter

The incenter is the point where all three internal angle bisectors of a triangle intersect. The central angle formed by two bisectors is 90° + (∠Opposite Vertex) / 2.

Circumcenter

The intersection of the perpendicular bisectors of the sides. The angle subtended by an arc at the circumcenter is double the angle subtended at any point on the circle (∠BOC = 2 × ∠A).

Orthocenter

The point where the altitudes of the triangle meet. The sum of the angle at the orthocenter and the opposite vertex angle is always 180° (∠BHC + ∠A = 180°).

Excenter

The intersection of the external bisectors of two angles and the internal bisector of the third angle. The angle formed by the external bisectors is 90° − (∠Opposite Vertex) / 2.

Shortcut Trick

In quadrilateral ABCD, we are given AB is parallel to CD and AB = CD.

Since the alternate interior angles formed by transversal AC are equal, ∠BAC = ∠DCA.

For ΔABC and ΔCDA, we evaluate the known equal parts:

⇒ AB = CD (Given) | ∠BAC = ∠DCA (Alternate angles) | AC = CA (Common side)

Since two sides and their included angle are equal, the triangles are congruent by the Side-Angle-Side rule.

∴ The correct answer is ΔABC ≅ ΔCDA by SAS.

Alternate Method

Given:

⇒ Quadrilateral ABCD with diagonal AC.

⇒ AB is parallel to CD.

⇒ AB = CD.

Concept Used:

⇒ If two parallel lines are intersected by a transversal, their alternate interior angles are equal.

⇒ SAS Congruence Rule: If two sides and the included angle of one triangle are equal to the corresponding parts of another triangle, they are congruent.

A B C D

AB = x CD = x

Calculations:

⇒ Consider ΔABC and ΔCDA.

⇒ Since AB is parallel to CD and AC acts as the transversal, the alternate interior angles are exactly equal.

⇒ ∠BAC = ∠DCA

⇒ We are already given that side AB = side CD.

⇒ The diagonal AC is a common side to both triangles, so AC = CA.

⇒ We now have two sides and the included angle equal for both triangles.

⇒ Therefore, ΔABC ≅ ΔCDA by the SAS congruence criterion.

∴ The correct answer is ΔABC ≅ ΔCDA by SAS.

Additional Information

SAS (Side-Angle-Side) Congruence

Two triangles are congruent if two sides and the included angle of one are identically equal to the corresponding parts of the other.

SSS (Side-Side-Side) Congruence

Two triangles are completely congruent if all three sides of one triangle are equal to the corresponding three sides of the other.

ASA (Angle-Side-Angle) Congruence

If two angles and the included side of one triangle are equal to the corresponding parts of another triangle, they are completely congruent.

Shortcut Trick

The centroid of a triangle divides the median in the ratio 2 : 1 from the vertex, so the ratio of lengths AG : AM is 2 : 3.

Since line DE is parallel to base BC, ΔADE and ΔABC are similar triangles.

The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides or medians: (2/3)² = 4/9.

This implies that if the area of ΔADE is 4 units, the area of the entire ΔABC is 9 units.

The area of trapezoid DECB = 9 − 4 = 5 units.

∴ The correct answer is 4:5.

Alternate Method

Given:

Line DE is parallel to side BC.

DE passes through the centroid G of ΔABC.

Formula Used:

For two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding heights or medians.

Area(ΔADE) ÷ Area(ΔABC) = (AG ÷ AM)²

2 parts 1 part

A B C D E M G

Area = 4 units Area = 5 units

Calculations:

⇒ Let AM be the median to the side BC. Since G is the centroid, it divides the median AM in the ratio 2 : 1.

⇒ AG : GM = 2 : 1

⇒ AG ÷ AM = AG ÷ (AG + GM) = 2 ÷ (2 + 1) = 2 / 3

⇒ In ΔADE and ΔABC, since DE || BC, their corresponding angles are equal, meaning ΔADE ∼ ΔABC.

⇒ Ratio of Area(ΔADE) to Area(ΔABC) = (AG ÷ AM)² = (2 / 3)² = 4 / 9

⇒ Let the area of ΔADE be 4x and the area of ΔABC be 9x.

⇒ The area of trapezoid DECB = Area(ΔABC) − Area(ΔADE) = 9x − 4x = 5x.

⇒ The required ratio = Area(ΔADE) : Area(DECB) = 4x : 5x = 4 : 5.

∴ The correct answer is 4:5.

Additional Information

Centroid of a Triangle

The centroid is the point of intersection of the three medians of a triangle. It always divides each median into a 2 : 1 ratio, with the longer part being closest to the vertex.

Similarity Area Theorem

If two triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides, altitudes, or medians.

Basic Proportionality Theorem (Thales Theorem)

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

Shortcut Trick

The length of an internal (transverse) common tangent is given directly by the formula: √[d² − (r₁ + r₂)²]

Here, d = 17, r₁ = 9, and r₂ = 5.

Length = √[17² − (9 + 5)²]

Length = √[289 − 196] = √93 cm

∴ The correct answer is √93 cm.

Alternate Method

d = 17 cm

9 cm 5 cm

Internal Common Tangent

Given:

⇒ Radius of the first circle (r₁) = 9 cm

⇒ Radius of the second circle (r₂) = 5 cm

⇒ Distance between their centers (d) = 17 cm

Formula Used:

⇒ Length of internal common tangent = √[d² − (r₁ + r₂)²]

Calculations:

⇒ Length = √[17² − (9 + 5)²]

⇒ Length = √[17² − 14²]

⇒ Length = √[289 − 196]

⇒ Length = √93 cm

∴ The correct answer is √93 cm.

Additional Information

Internal (Transverse) Common Tangent

It is the tangent that intersects the line segment joining the centers of the two circles. Formula: TCT = √[d² − (r₁ + r₂)²]

External (Direct) Common Tangent

It is the tangent that does not intersect the line segment joining the centers. Formula: DCT = √[d² − (r₁ − r₂)²]

Number of Tangents Rule

Two non-intersecting circles can have a maximum of 4 common tangents (2 direct and 2 transverse). If they touch externally, they have exactly 3 common tangents.

Shortcut Trick For two circles with radii r1 and r2, and distance between centers d, the formula is: Transverse Common Tangent = √(d2 − (r1 + r2)2) TCT = √(202 − (7 + 3)2) TCT = √(400 − 100) = √300 cm ∴ The correct answer is √300 cm.

Alternate Method

C1

C2 r₁=7 r₂=3

d = 20 cm T1 T2

TCT

Given: Radius of first circle (r1) = 7 cm Radius of second circle (r2) = 3 cm Distance between centers (d) = 20 cm Formula Used: Length of Transverse Common Tangent (TCT) = √(d2 − (r1 + r2)2) Calculations: ⇒ TCT = √(202 − (7 + 3)2) ⇒ TCT = √(202 − 102) ⇒ TCT = √(400 − 100) ⇒ TCT = √300 cm ∴ The correct answer is √300 cm.

Additional Information Direct Common Tangent (DCT) The formula for the length of a Direct Common Tangent is: DCT = √(d2 − (r1 − r2)2). Number of Common Tangents If two circles are strictly apart (d > r1 + r2), they have exactly 4 common tangents (2 direct and 2 transverse). Intersection Property The Transverse Common Tangent crosses the line joining the centers of the two circles, dividing the distance internally exactly in the ratio of their radii (r1 : r2).

Shortcut Trick According to circle properties, a shorter chord is always farther away from the center compared to a longer chord. The given chord lengths are 9 cm and 7 cm. Since 7 cm < 9 cm, the 7 cm chord must be farther from the center than the 9 cm chord. The 9 cm chord is at a distance of 4 cm. Therefore, the distance of the 7 cm chord from the center must be greater than 4 cm. ∴ The correct answer is greater than 4 cm.

Alternate Method Given: Length of the first chord = 9 cm Distance of the first chord from center (d1) = 4 cm Length of the second chord = 7 cm Formula Used: By Pythagoras theorem in a right-angled triangle formed by the radius, distance to the chord, and half of the chord: r2 = d2 + (L ÷ 2)2

O

A B 9 cm

C D 7 cm

4 cm d = ?

r r

Calculations: ⇒ For the 9 cm chord: r2 = 42 + (9 ÷ 2)2 ⇒ r2 = 16 + 4.52 ⇒ r2 = 16 + 20.25 = 36.25 ⇒ For the 7 cm chord: r2 = d2 + (7 ÷ 2)2 ⇒ 36.25 = d2 + 3.52 ⇒ 36.25 = d2 + 12.25 ⇒ d2 = 36.25 − 12.25 ⇒ d2 = 24 ⇒ d = √24 = 2√6 ≈ 4.89 cm ⇒ Since 4.89 cm is strictly greater than 4 cm, the distance is greater than 4 cm. ∴ The correct answer is greater than 4 cm.

Additional Information Distance of Chords from Center Of any two chords of a circle, the larger chord is nearer to the center, and the shorter chord is farther from the center. Perpendicular from Center The perpendicular drawn from the center of a circle to a chord always bisects the chord into two equal segments. Equal Chords Property Chords that are equal in length are precisely equidistant from the center of the circle.

Shortcut Trick

By the extended Sine Rule for any circumscribed triangle:

Side / sin(θ) = 2 × Circumradius (r)

Let the required side opposite to the 120° angle be x.

⇒ x / sin(120°) = 2r

⇒ x = 2r × (√3 / 2) = r√3

∴ The correct answer is r√3.

Alternate Method

Given:

Angle of the inscribed triangle, ∠A = 120°

Radius of the circumscribed circle = r

Formula Used:

Opposite angles of a cyclic quadrilateral sum to 180°

Angle at the center = 2 × Angle at the circumference

sin(θ) = Perpendicular / Hypotenuse

O

120° B C A M

r 60°

BC = ?

Calculations:

⇒ Let the required side opposite to the 120° angle be chord BC, and A be the vertex containing the 120° angle.

⇒ Let D be an imaginary point on the major arc. ABDC forms a cyclic quadrilateral.

⇒ Opposite angles sum to 180° ⇒ ∠BDC = 180° − 120° = 60°.

⇒ The angle subtended by chord BC at the center O is ∠BOC = 2 × ∠BDC = 2 × 60° = 120°.

⇒ Draw a perpendicular OM from center O to BC. It bisects both ∠BOC and chord BC.

⇒ In right-angled ΔOMB, ∠BOM = 120° ÷ 2 = 60°.

⇒ sin(60°) = BM / OB

⇒ √3 / 2 = BM / r ⇒ BM = r√3 / 2

⇒ Length of side BC = 2 × BM = 2 × (r√3 / 2) = r√3

∴ The correct answer is r√3.

Additional Information

Sine Rule (Extended Law of Sines)

For any triangle inscribed in a circle of radius R, the ratio of any side to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C) = 2R.

Angle Subtended by Arc

The angle subtended by an arc at the center is double the angle subtended by it at any remaining part of the circle.

Cyclic Quadrilateral Property

In a cyclic quadrilateral, the sum of opposite interior angles is always 180°. This property helps connect major and minor arc angles.

Shortcut Trick

Convert the standard fractions into their known decimal values directly.

1/8 = 0.125

1/16 = 0.0625

Sum = 0.125 + 0.0625 = 0.1875

∴ The correct answer is 0.1875.

Alternate Method

Given:

Expression = 1/8 + 1/16

Calculation:

⇒ Find the Least Common Multiple (LCM) of 8 and 16, which is 16.

⇒ Multiply numerator and denominator of 1/8 by 2 to make denominators equal.

⇒ Expression = (1 × 2)/(8 × 2) + 1/16

⇒ Expression = 2/16 + 1/16

⇒ Expression = 3/16

⇒ Now, convert the fraction into a decimal by dividing 3 by 16.

⇒ 3 ÷ 16 = 0.1875

∴ The correct answer is 0.1875.

Additional Information

Common Fraction to Decimal Conversions

1/2 = 0.5 | 1/4 = 0.25 | 1/8 = 0.125 | 1/16 = 0.0625. Memorising these standard fraction equivalents significantly speeds up calculations.

Addition of Decimals

Always align the decimal points vertically before adding. You can add trailing zeros to make all numbers have the same number of decimal places (e.g., 0.1250 + 0.0625 = 0.1875).

Simplification Rules

While solving arithmetic expressions involving fractions, always equalise the denominators first by taking their LCM before adding or subtracting the numerators.

The correct answer is: Stubborn (Option 3).

Key Points

• The word "Intransigent" means unwilling or refusing to change one's views or to agree about something. (अडिग/हठी)
• Example: The intransigent manager refused to negotiate with the striking employees.
• "Stubborn" refers to a person's determination not to change their mind or attitude, no matter the situation or external pressure. (ज़िद्दी)
• Example: The stubborn child refused to eat anything other than pizza for dinner.
• Hence, we can infer that the synonym of 'Intransigent' is 'Stubborn'.

Therefore, the correct answer is: Stubborn (Option 3).

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Willing (इच्छुक): Ready, eager, or prepared to do something.
• Example: She was willing to help her friend with the project despite her busy schedule.
• Compliant (अनुकूल): Inclined to agree with others or obey rules, especially to an excessive degree.
• Example: The compliant employee followed every company policy without question.
• Submissive (आज्ञाकारी): Ready to conform to the authority or will of others; meekly obedient or passive.
• Example: The dog was submissive to its owner's commands during training sessions.

The correct answer is: Option 2 (Ugliness).

Key Points

• The word "Pulchritude" means physical beauty, especially that of a woman. (सौंदर्य)
• Example: The actress was admired for her pulchritude and grace, which made her a favorite among directors.
• "Ugliness" is the opposite of pulchritude and refers to the quality of being unpleasant or unattractive in appearance. (कुरूपता)
• Example: The old building's ugliness was evident in its peeling paint and broken windows.
• Hence, the most appropriate antonym of "Pulchritude" is "Ugliness."

Therefore, the correct answer is: Option 2 (Ugliness).

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Grace (अनुग्रह): Elegance or beauty of movement, form, or expression.
• Example: She danced with such grace that the audience was mesmerized.
• Charm (आकर्षण): The power or quality of delighting, attracting, or fascinating others.
• Example: His sense of humor added to his natural charm.
• Beauty (सुंदरता): A combination of qualities, such as shape, color, or form, that pleases the aesthetic senses.
• Example: The beauty of the mountain landscape left everyone in awe.

The correct answer is: Option 2, i.e., "In a state of depression or inactivity".

Key Points

• The phrase "In the doldrums" means being in a state of depression or inactivity, where things seem stagnant and unprogressive. (निराशा या निष्क्रियता की स्थिति में होना)
• Example: After the company lost its biggest client, it has been in the doldrums for months, struggling to recover.
• Explanation of the phrase: The term "doldrums" originates from maritime terminology, referring to parts of the ocean near the equator known for calm winds, where ships would often get stuck. Figuratively, it describes a state of stagnation or low energy. (समुद्र की स्थिति को दर्शाने वाले शब्द से लिया गया है, जहां जहाज रुक जाते थे, और इसे अब स्थिरता या कम ऊर्जा की स्थिति के लिए इस्तेमाल किया जाता है।)
• Example: John felt in the doldrums after failing his exams, but his friends motivated him to try again.

Therefore, the correct answer is: Option 2, i.e., "In a state of depression or inactivity".

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Option 1: Sailing in high winds (तेज हवाओं में नौकायन): Refers to navigating a boat in strong winds, often requiring skill and effort.
• Example: The sailors were sailing in high winds to reach the shore before the storm hit.
• Option 3: On the verge of success (सफलता के कगार पर): Close to achieving success or accomplishing a goal.
• Example: After years of hard work, she was finally on the verge of success in her career.
• Option 4: Working with great energy (बहुत ऊर्जा के साथ काम करना): Describes someone putting in a lot of effort and enthusiasm into their tasks.
• Example: The team was working with great energy to meet the project deadline.

The correct answer is: Option 4 (Reclaimable).

Key Points

• The word "Incorrigible" refers to someone or something that is incapable of being corrected, improved, or reformed, often due to deeply ingrained habits. (असुधारनीय)
• Example: Despite numerous attempts to change his behavior, he remains an incorrigible liar.
• "Reclaimable" refers to something that can be recovered, restored, or rehabilitated. It is the opposite of incorrigible, as it implies the potential for improvement or change. (पुनः प्राप्त करने योग्य)
• Example: The damaged land was considered reclaimable after extensive environmental efforts.
• Hence, the most appropriate antonym of ‘Incorrigible’ is ‘Reclaimable’.

Therefore, the correct answer is: Option 4 (Reclaimable).

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Chronic (दीर्घकालिक): Refers to something that persists for a long time or constantly recurs.
• Example: She suffers from chronic back pain that requires regular treatment.
• Deep-rooted (गहराई से जुड़ा हुआ): Indicates something firmly established or ingrained over time.
• Example: The community's deep-rooted traditions are passed down through generations.
• Habitual (आदतन): Refers to something done regularly or as a habit.
• Example: He is a habitual latecomer to meetings.

The correct answer is option 3.

Key Points

The correct spelling of the drug-related word is 'hallucinogenic'.

• Option 3 'Hallucinogenic' is the correct spelling.
• The word 'hallucinogenic' is an adjective that refers to something capable of causing hallucinations.
• For Example - Certain mushrooms have hallucinogenic properties.

Therefore, the correct answer is 'Hallucinogenic'.

Additional Information

Let's understand the incorrect options - 

• Option 1 'Hallucinagenic' - Incorrect spelling, as it uses 'a' instead of 'o' in the middle.
• Option 2 'Hallucinognic' - Incorrect spelling, as it omits the 'e' after 'g'.
• Option 4 'Halluscinogenic' - Incorrect spelling, as it adds an unnecessary 's' after 'u'.

The correct answer is: Option 4 (Apostate).

Key Points

• The word "Apostate" refers to a person who renounces or abandons their religious faith, principles, or allegiance. (धर्मत्यागी)
• Example: He was labeled an apostate after he publicly declared that he no longer followed the teachings of his religion.
• "Renegade" is often used to describe someone who betrays or deserts a cause, organization, or set of principles, but it does not specifically imply abandoning religious faith. (देशद्रोही)
• Example: The renegade soldier refused to follow orders and joined the opposition forces.
• Hence, the correct one-word substitution for "One who abandons his religious faith" is "Apostate."

Therefore, the correct answer is: Option 4 (Apostate).

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Renegade (देशद्रोही): Someone who deserts or betrays an organization, country, or set of principles.
• Example: The renegade politician turned against his own party to support the opposition.
• Atheist (नास्तिक): A person who does not believe in the existence of God or gods.
• Example: She identifies as an atheist and does not follow any religious practices.
• Heretic (विधर्मी): Someone who holds beliefs or opinions that go against the established doctrines of a religion, especially Christianity.
• Example: In the medieval period, heretics were often punished for their beliefs.

The correct answer is: Schism.

Key Points

• The word "Schism" refers to a formal division or split within a religious organization, typically due to differences in opinion or belief. (मतभेद के कारण धार्मिक संगठन में औपचारिक विभाजन)
• Example: The Great Schism of 1054 divided the Christian Church into the Eastern Orthodox and Roman Catholic branches.
• "Schism" is the correct term for a separation or division within a group, especially a religious one, due to conflicting views. (विभाजन)
• Example: A schism occurred within the congregation when they could not agree on the new policies.

Therefore, the correct answer is: Schism.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Sedition (राजद्रोह): Conduct or speech inciting people to rebel against the authority of a state or monarch.
• Example: The activists were arrested on charges of sedition for their anti-government speeches.
• Heresy (कुशिक्षा): A belief or opinion that goes against established religious doctrines.
• Example: During the Middle Ages, individuals accused of heresy were often severely punished.
• Apostasy (धर्मत्याग): The abandonment or renunciation of a religious or political belief.
• Example: His apostasy shocked the community when he publicly declared his rejection of the faith.

The correct answer is 'has there been'.

 

Key Points

• The sentence is structured as an example of inversion, which is used for emphasis, especially with negative adverbs such as "Rarely," "Never," or "Seldom."
• In such cases, the auxiliary verb comes before the subject.
• Option 1: "has there been" correctly follows this inversion rule and is grammatically accurate.
• Options 2, 3, and 4 do not follow the correct structure or tense required in the sentence, making them incorrect.

Therefore, the correct answer is Option 1.

 

Complete Sentence: "Rarely has there been such an overwhelming consensus among critics."

 

Additional Information

• Option 2: "there has been" is incorrect because it does not follow the inversion rule required after "Rarely."
• Option 3: "was there" is incorrect as it changes the tense unnecessarily and does not align with the context.
• Option 4: "had been" is incorrect as it uses the past perfect tense, which is not required here.

The correct answer is 'reconcile'.

 

Key Points

• The verb "reconcile" means to bring together or make consistent conflicting ideas, accounts, or facts.
• In this sentence, the historian had to resolve or harmonize the contradictory accounts of the regime's collapse, which aligns with the meaning of "reconcile."
• Options 1, 2, and 4 are incorrect because they do not convey the meaning of resolving or harmonizing contradictions.

Therefore, the correct answer is Option 3.

 

Complete Sentence: "Only after years of archival research did the historian manage to reconcile the contradictory accounts of the regime's collapse."

 

Additional Information

• Option 1: "retract" is incorrect because it means to withdraw or take back something, which does not fit the context.
• Option 2: "revise" is incorrect because it means to change or modify something, but it does not convey the idea of harmonizing contradictions.
• Option 4: "reverberate" is incorrect because it means to echo or resound, which is unrelated to the context.